Derivative of \( \displaystyle \sin^{4 x - 3}{\left(4 x - 3 \right)} \)
Problem 2.834 · hard
Differentiate \( \displaystyle f(x) = \sin^{4 x - 3}{\left(4 x - 3 \right)} \).
- \[ \frac{d}{d x} \sin^{4 x - 3}{\left(4 x - 3 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]rewriteRewrite the power using the exponential identity.≈ Checked numerically
- \[ = e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \frac{d}{d x} \left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\left(4 x - 3\right) \frac{d}{d x} \ln{\left(\sin{\left(4 x - 3 \right)} \right)} + \ln{\left(\sin{\left(4 x - 3 \right)} \right)} \frac{d}{d x} \left(4 x - 3\right)\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(\left(4 x - 3\right) \frac{d}{d x} \ln{\left(\sin{\left(4 x - 3 \right)} \right)} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(\frac{\left(4 x - 3\right) \frac{d}{d x} \sin{\left(4 x - 3 \right)}}{\sin{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{4 \left(4 x - 3\right) \cos{\left(4 x - 3 \right)}}{\sin{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]derivativeDifferentiate the sine function.✓ Proved
- \[ = \left(\frac{\left(16 x - 12\right) \cos{\left(4 x - 3 \right)}}{\sin{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \]algebraSubstitute the original function back for the exponential term.≈ Checked numerically
- \[ = \left(\left(16 x - 12\right) \cot{\left(4 x - 3 \right)} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \]simplifySimplify the expression using the cotangent identity.≈ Checked numerically
- \[ = 4 \left(\left(4 x - 3\right) \cot{\left(4 x - 3 \right)} + \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \]simplifyFactor out the common constant 4.✓ Proved
Answer \( \left(\frac{16 x - 12}{\tan{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \)
✓ Nihil obstat Lines: 8 proved, 3 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*(((3 - 4*x)*cos(4*x - 3) - log(sin(4*x - 3))*sin(4*x - 3))*exp((4*x - 3)*log(sin(4*x - 3))) + ((4*x - 3)*cos(4*x - 3) + log(sin(4*x - 3))*sin(4*x - 3))*sin(4*x - 3)**(4*x - 3))/sin(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*(((3 - 4*x)*cos(4*x - 3) - log(sin(4*x - 3))*sin(4*x - 3))*sin(4*x - 3)**(4*x - 3) + ((4*x - 3)*cos(4*x - 3) + log(sin(4*x - 3))*sin(4*x - 3))*exp((4*x - 3)*log(sin(4*x - 3))))/sin(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*(((3 - 4*x)*cot(4*x - 3) - log(sin(4*x - 3)))*sin(4*x - 3)**(4*x - 2) + ((4*x - 3)*cos(4*x - 3) + log(sin(4*x - 3))*sin(4*x - 3))*sin(4*x - 3)**(4*x - 3))/sin(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 cot has poles at multiples of pi |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments cot has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(4*x - 3) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies logarithmic differentiation via rewriting as an exponential. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies logarithmic differentiation via exponential rewriting. Each step isolates a single rule application (rewrite, chain, product, derivative, simplify) and uses labels from the allowed vocabulary. The final answer matches the stated answer.gpt-oss:20b: pass 2026-09-26
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-27 with SymPy 1.14.0.